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Rational Jacobi elliptic solutions for nonlinear differential-difference lattice equations. (English) Zbl 1252.34013

Summary: We present a direct new method for constructing the rational Jacobi elliptic solutions for nonlinear differential-difference equations, which may be called the rational Jacobi elliptic function method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential-difference equations in mathematical physics via the lattice equation. The proposed method is effective for obtaining the exact solutions for nonlinear differential-difference equations.

MSC:

34A33 Ordinary lattice differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
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[1] Fermi, E.; Pasta, J.; Ulam, S., Collected papers of enrico Fermi II, (1965), University of Chicago Press Chicago, IL
[2] Su, W.P.; Schrieffer, J.R.; Heeger, A.J., Solitons in polyacetylene, Phys. rev. lett., 42, 1698-1701, (1979)
[3] Davydov, A.S., The theory of contraction of proteins under their excitation, J. theoret. biol., 38, 559-569, (1973)
[4] Marquié, P.; Bilbault, J.M.; Remoissenet, M., Observation of nonlinear localized modes in an electrical lattice, Phys. rev. E, 51, 6127-6133, (1995)
[5] Toda, M., Theory of nonlinear lattices, (1989), Springer Berlin · Zbl 0694.70001
[6] Wadati, M., Transformation theories for nonlinear discrete systems, Prog. suppl. theor. phys., 59, 36-63, (1976)
[7] Ohta, Y.; Hirota, R., A discrete KdV equation and its Casorati determinant solution, J. phys. soc. Japan, 60, 2095, (1991)
[8] Ablowitz, M.J.; Ladik, J., Nonlinear differential – difference equations, J. math. phys., 16, 598-603, (1975) · Zbl 0296.34062
[9] Hu, X.B.; Ma, W.X., Application of hirota’s bilinear formalism to the Toeplitz lattice some special soliton-like solutions, Phys. lett. A, 293, 161-165, (2002) · Zbl 0985.35072
[10] Baldwin, D.; Goktas, U.; Hereman, W., Symbolic computation of hyperbolic tangent solutions for nonlinear differential – difference equations, Comput. phys. commun., 162, 203-217, (2004) · Zbl 1196.68324
[11] Liu, S.K.; Fu, Z.T.; Wang, Z.G.; Liu, S.D., Periodic solutions for a class of nonlinear differential – difference equations, Commun. theor. phys., 49, 1155-1158, (2008) · Zbl 1392.34085
[12] Qiong, C.; Bin, L., Application of Jacobi elliptic function expansion method for nonlinear difference equations, Commun. theor. phys., 43, 385-388, (2005)
[13] Xie, F.; Jia, M.; Zhao, H., Some solutions of discrete sine – gordon equation, Chaos solitons fractals, 33, 1791-1795, (2007) · Zbl 1129.35456
[14] Zhu, S.D., Exp-function method for the hybrid-lattice system, Int. J. nonlinear sci., 8, 461, (2007)
[15] Aslan, I., A discrete generalization of the extended simplest equation method, Commun. nonlinear sci. numer. simul., 15, 1967-1973, (2010) · Zbl 1222.65114
[16] Yang, P.; Chen, Y.; Li, Z.B., ADM-padé technique for the nonlinear lattice equations, Appl. math. comput., 210, 362-375, (2009) · Zbl 1162.65399
[17] Zhu, S.D.; Chu, Y.M.; Qiu, S.L., The homotopy perturbation method for discontinued problems arising in nanotechnology, Comput. math. appl., 58, (2009) · Zbl 1189.65186
[18] Zhang, S.; Dong, L.; Ba, J.; Sun, Y., The (G’/G)-expansion method for nonlinear differential – difference equations, Phys. lett. A, 373, 905-910, (2009) · Zbl 1228.34096
[19] Aslan, I., The ablowitz – ladik lattice system by means of the extended \((G^\prime / G)\)-expansion method, Appl. math. comput., 216, 2778-2782, (2010) · Zbl 1193.35179
[20] Zhang, S., Discrete Jacobi elliptic function expansion method for nonlinear difference equation, Phys. scr., 80, 045002-045010, (2009)
[21] Gepreel, Khaled A., Rational Jacobi elliptic solutions for nonlinear difference differential equations, Nonlinear sci. lett. A, 2, 151-158, (2011)
[22] Wu, G.; Xia, T., A new method for constructing soliton solutions to differential – difference equation with symbolic computation, Chaos solitons fractals, 39, 2245-2248, (2009) · Zbl 1197.35250
[23] Xie, F.; Wang, J., A new method for solving nonlinear differential – difference equation, Chaos solitons fractals, 27, 1067-1071, (2006) · Zbl 1094.34058
[24] Liu, C., Exponential function rational expansion method for nonlinear differential – difference equations, Chaos solitons fractals, 40, 708-716, (2009) · Zbl 1197.35243
[25] Wang, Q.; Yu, Y., New rational formal for (1+1)-dimensional Toda equation and another Toda equation, Chaos solitons fractals, 29, 904-915, (2006) · Zbl 1142.37370
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